With the advent of computer technology, various methods of synthesizing and analyzing patterns (e.g., images and waveforms, such as sounds and various electromagnetic waveforms including light, electricity, etc.) have been developed. Various techniques for synthesizing images are used, for example, in various computer graphics programs and in a variety of other applications—such as computer screen saver programs and a wide range of other applications. In addition, various techniques for analyzing existing images have also been developed in the existing art.
While a variety of techniques are known, there still remains a great need for improvements in pattern synthesis and in pattern analysis. Although computers are becoming more versatile and intelligent, there remains are great need to simplify functions and operations within computers to save valuable memory space and to enable quick and accurate determinations to be performed by the computers. In the arts of pattern synthesis and pattern analysis, a variety of mathematical concepts have been put forth in an effort to simplify pattern synthesis and analysis. However, while there have been improvements in prior methods, there remains a need for fundamental improvements in such methods.
While variation in form and shape has always intrigued students of nature (e.g., such as biologists and mathematicians), creating accurate and simple characterizations of form and shape has proven to be a conceptually arduous task. There remains a need for simple mathematical and biophysical rules underlying shapes in general and morphology and morphogenesis.
The ancient Greeks developed some basic geometric principles to explain natural forms. In both ancient and modern concepts, the circle prevails as the ideal object. Generally, circular and cylindrical forms and shapes can also be observed in plants and organisms. Much of existing geometry is based on the circle, including all trigonometry and all technology based on trigonometric functions. Complex forms can also be analyzed in terms of circles and harmonics.
More recently, other findings to describe natural forms have included, for example, fractals and algorithms that can, for example, generate some types of virtual plants.
With the advancement of computer technology, models have become very sophisticated. In such models, mathematics has been used as a tool. But these tools alone have not unraveled principles underlying shapes. Moreover, natural forms in general fail to follow exact mathematical rules: e.g., perfect circles are never observed in nature; and no two single leaves are ever exactly the same. With existing algorithms it remains impossible to describe forms as they exist in nature and it remains impossible to conceptualize the true relationships between various shapes and forms.
There remains a need for those in the applied mathematical and biological fields to be able to characterize various shapes and forms utilizing ever simpler geometrical rules. Similarly, there remains a need in the arts of pattern synthesis and analysis for improved methods whereby a wide variety of patterns (e.g., geometric shapes, waveforms, etc.) can be quickly and accurately synthesized and/or analyzed with optimal use of computer memory and resources.